Finding Pareto Optimal Solution

Evolutionary algorithms are correctly fitted to multiobjective optimization problems as they are essentially based on biological processes, which is inherently multiobjective. An extensive survey on multiobjective evolutionary algorithms are well presented in [99]. Central to these articles, considering superior performance

for solving multi objective problems, the 2LB-MOPSO [94] and NSGA-II [100] algorithms have been used in this study.

In NSGA-II, initially a random population of size H, which is sorted based on the non-domination, is created. This population subsequently undergoes selection, crossover and mutation processes to produce an offspring population of size H. A combined population of size 2H is formed from the parent and offspring population. Next, the population is sorted according to the non-domination relation. This in turn classifies the complete population into several non-dominated fronts based on the values of the objective functions. Until each member of the population falls into one front, the other fronts are determined. The new parent population is generated by adding the solutions from the first front. Several non-dominated fronts are discarded as the population size is predefined. The required numbers of members for the new population are selected using a new parameter called crowding distance. The crowding distance describes how close an individual is to its neighbors.

Similar to GA, the PSO algorithm has been successfully extended to multiobjective optimization problems. Different from other variants of MOPSO algorithms, the 2LB-MOPSO algorithm uses two local bests instead of one personal best and one global best to lead each particle. The two local bests are selected to be close to each other in order to enhance the local search ability of the algorithm. Compared to the other variants of MOPSO algorithms, the 2LB-MOPSO algorithm shows great advantages in maintaining a good diversity of the solutions, convergence speed and fine-searching ability.

In 2LB-MOPSO algorithm, N P number of particles are randomly and uniformly initialized in the D-dimensional search space. Next, the fitness values of all particles are evaluated and all current positions set to be A(0), the external archive. An external archive is commonly used to store the non-dominated solutions obtained in the search process. The size of the archive is usually restricted to a pre-specified size which is normally the same as the finally required approximation solution set size. In 2LB-MOPSO algorithm, the initialized archive includes all initialized solutions at iteration 1. In order to select the first lbests for a particle from the A(0), an objective is first randomly selected followed by a

random selection of a non empty bin of the chosen objective. Within this bin, the archived member with the lowest front number and among these with the highest crowding distance is selected as the first lbests. The second lbests is selected from a neighboring non empty bin with the lowest front number and the smallest Euclidean distance in the parameter space to the first lbests. As velocity of each particle is adjusted by the two lbests from two neighboring bins, the flight of each particle will be in the direction between the positions of two lbests and oriented to improve upon the current non-dominated solutions.

Since the first lbests of every particle is chosen randomly, every particle should not be assigned with a new pair of lbests which come from the different pair of bins in every iteration. This is because the flight of each particle will be almost random in this case. Therefore, after assigning a pair of lbests to a particle, the number of iterations the particle fails to contribute a solution to the archive A(t) is counted. The particle is reassigned with another pair of lbests when the count exceeds a pre-specified threshold. When the count is less than or equal to the pre-specified threshold during the iterative optimization stage, two lbests are chosen from the same assignment of the objective and the bin as used in the last iteration. The particle will accelerate potentially in a direction between the two lbests and hence may explore the region of the two lbests.

The velocity and position of each particle are updated. If any dimension exceeds the search space, then they are reset to the corresponding bound value. Next, the fitness value of a particle is evaluated. In every iteration, all new positions Q(t) generated in iteration t is combined with the members in the archive

A(t) to obtain the mixed-temporary external archive. The sorted archive R(t) is

obtained by applying the non-domination sorting to this mixed-temporary archive. During this process, all the sorted solutions retain two indicators, namely, the front rank and crowding distance value. The sorted solution with the lowest front rank is first included in the archive A(t + 1). When the size of the archive equals to the permitted maximum size of the archive, the crowding distance is applied to select the required number of members to be included in A(t + 1) from the lowest front that still remains unselected in the archive R(t).